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Properties

Label 16.12.1571275555...5489.1
Degree $16$
Signature $[12, 2]$
Discriminant $23^{10}\cdot 41^{14}$
Root discriminant $182.92$
Ramified primes $23, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 16T817

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-49226183, 361090269, -370583547, 18095388, 98409063, -19074130, -10886657, 1934875, 918855, -22955, -74604, -5900, 3957, 296, -103, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 103*x^14 + 296*x^13 + 3957*x^12 - 5900*x^11 - 74604*x^10 - 22955*x^9 + 918855*x^8 + 1934875*x^7 - 10886657*x^6 - 19074130*x^5 + 98409063*x^4 + 18095388*x^3 - 370583547*x^2 + 361090269*x - 49226183)
 
gp: K = bnfinit(x^16 - 4*x^15 - 103*x^14 + 296*x^13 + 3957*x^12 - 5900*x^11 - 74604*x^10 - 22955*x^9 + 918855*x^8 + 1934875*x^7 - 10886657*x^6 - 19074130*x^5 + 98409063*x^4 + 18095388*x^3 - 370583547*x^2 + 361090269*x - 49226183, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 103 x^{14} + 296 x^{13} + 3957 x^{12} - 5900 x^{11} - 74604 x^{10} - 22955 x^{9} + 918855 x^{8} + 1934875 x^{7} - 10886657 x^{6} - 19074130 x^{5} + 98409063 x^{4} + 18095388 x^{3} - 370583547 x^{2} + 361090269 x - 49226183 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1571275555715210001755383712793895489=23^{10}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $182.92$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $23, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{187100792013042265763267302364004728030327407462882772} a^{15} - \frac{17649148933001391529905569903799186155348798653962871}{187100792013042265763267302364004728030327407462882772} a^{14} - \frac{20105034834544636613930438988448109673296182564704205}{187100792013042265763267302364004728030327407462882772} a^{13} + \frac{10027918469867476542888957519438821044366882187579779}{93550396006521132881633651182002364015163703731441386} a^{12} + \frac{41940369405007985361233119484310747664238538265716701}{187100792013042265763267302364004728030327407462882772} a^{11} - \frac{15326635530948472379610849920045288650201822782173879}{187100792013042265763267302364004728030327407462882772} a^{10} - \frac{30057518300374004983274536132382235171132355361448449}{187100792013042265763267302364004728030327407462882772} a^{9} + \frac{1928094110071103599862888245911437749646858074815373}{187100792013042265763267302364004728030327407462882772} a^{8} + \frac{47503404054817049970415193903407350156086135518620895}{187100792013042265763267302364004728030327407462882772} a^{7} - \frac{13621368454308609329593909400973943194476044708198529}{46775198003260566440816825591001182007581851865720693} a^{6} + \frac{20600210434640117160316045399992683644979509693948759}{46775198003260566440816825591001182007581851865720693} a^{5} - \frac{35750361552553867335067362823199212257459775839336219}{187100792013042265763267302364004728030327407462882772} a^{4} - \frac{36803750603351833141301828981268439731978675673318765}{187100792013042265763267302364004728030327407462882772} a^{3} - \frac{15562197952508424622496873596752846595074990852125255}{46775198003260566440816825591001182007581851865720693} a^{2} + \frac{3443858201397697771634776473015724074615060180667128}{46775198003260566440816825591001182007581851865720693} a - \frac{88655779332416004247676294497888118851940183125668561}{187100792013042265763267302364004728030327407462882772}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3719997282040 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T817:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 512
The 32 conjugacy class representatives for t16n817
Character table for t16n817 is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.68921.1, 8.8.54500230757132921.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{12}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$23$23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$